Summary
The behavior of a slightly rarefied gas mixture bounded by plane boundaries is investigated on the basis of the linearized Boltzmann equation ofB-G-K type for gas mixtures under the diffusive boundary condition. A useful result of the present analysis is that the macroscopic equations and the appropriate boundary conditions in terms of slip and jump are obtained together with the Knudsen-layer corrections near the boundaries. This system of equations makes possible the treatment at fluid dynamic level for various problems of gas mixtures with plane geometry which require kinetic theory consideration. As an application of this system, some basic flow problems of a slightly rarefied gas mixture, namely, Couette flow, thermal slip flow and diffusion slip flow between two plates are taken up. The total velocity distributions of these concrete problems are explicitly obtained for the first time, and their dependence on the properties and concentration of the component gases in the mixture are clarified in some detail.
Zusammenfassung
Das Verhalten einer verdünnten Gasmischung, bei kleiner aber nicht vernachlässigbarer Knudsen Zahl, zwischen zwei parallelen Platten wird analytisch untersucht. Die linearisierten Boltzmann Gleichungen desB-G-K Typs für Gasmischungen mit diffusiven Randbedingungen werden angewendet. Aus der vorliegenden Untersuchung resultieren brauchbare makroskopische Gleichungen mit den zugehörigen Randbedingungen, — mit Gleitgeschwindigkeit und Temperatursprung formuliert, — sowie die Korrekturen für die wandnahe Knudsen-Schicht. Verschiedene Strömungen von Gasmischungen, bei denen gas-kinetische Effekte eine Rolle spielen, entlang ebener Begrenzungen können mit den Gleichungssystemen auf strömungsmechanischem Niveau behandelt werden. Das System wird auf die Couette Strömung und die „thermal-slip“ und „diffusion-slip“ Strömungen angewendet. Zum ersten Mal werden die Geschwindigkeitsprofile dieser elementaren Strömungen explizit berechnet. Der Einfluß der Gaseigenschaften und Konzentrationen auf diese Profile werden weitgehend erklärt.
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References
Y. Onishi, inRarefied Gas Dynamics, edited by H. Oguchi (Univ. of Tokyo Press, Tokyo, 1984), Vol. II, pp. 875–884.
B. B. Hamel, Phys. of Fluids8, 418 (1965).
V. M. Zhdanov, Sov. Phys.-Tech. Phys.12, 134 (1967).
B. K. Annis and E. A. Mason, Phys. of Fluids13, 1452 (1970).
S. K. Loyalka, Phys. of Fluids14, 2599 (1971).
Yu. I. Yalamov, A. A. Yushkanov and M. N. Gaidukov, Sov. Phys.-Tech. Phys.20, 134 (1975).
H. A. Kramers and J. Kistemaker, Physica10, 699 (1943).
J. R. Brock, J. Colloid Sci.18, 489 (1963).
Yu. I. Yalamov, I. N. Ivchenko and B. V. Deryagin, Sov. Phys. Dokl.13, 446 (1968).
H. Lang and S. K. Loyalka, Phys. of Fluids13, 1871 (1970).
W. G. Vincenti and C. H. Kruger, Jr.,Introduction to Physical Gas Dynamics (John Wiley & Sons, New York, 1965), p. 384.
Y. Sone and Y. Onishi, J. Phys. Soc. Japan,44, 1981 (1978).
Y. Sone, inRarefied Gas Dynamics, edited by L. Trilling and H. Y. Wachman (Academic Press, New York, 1969), pp. 243–253.
Y. Sone, inRarefied Gas Dynamics, edited by D. Dini (Editrice Tecnico Scientifica, Pisa, Italy, 1971), Vol. II, p. 737.
C. Cercignani,Theory and Application of the Boltzmann Equation (Scottish Academic Press, Edinburgh and London, 1975), Chap. V.
Y. Sone, inRarefied Gas Dynamics, edited by H. Oguchi (Univ. of Tokyo Press, Tokyo, 1984), Vol. I, pp. 71–87.
M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (U.S. Department of Commerce, National Bureau of Standards, Applied Mathematics Series 55, 1964), p. 1001.
Y. Sone and Y. Onishi, J. Phys. Soc. Japan35, 1773 (1973).
L. Waldmann, inRarefied Gas Dynamics, edited by L. Talbot (Academic Press, New York, 1961), pp. 323–344.
L. Waldmann and K. H. Schmitt, inAerosol Science, edited by C. N. Davies (Academic Press, New York, 1966), pp. 137–162.
G. Baccaglini, D. R. Miller and P. A. Libby, Phys. of Fluids,14, 2299 (1971).
J. W. Cipolla, Jr., H. Lang and S. K. Loyalka, inRarefied Gas Dynamics, edited by M. Becker and M. Fiebig (DFVLR-Press, Porz-Wahn, Germany, 1974), pp. F.4-1–F.4-10.
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Onishi, Y. On the behavior of a slightly rarefied gas mixture over plane boundaries. Z. angew. Math. Phys. 37, 573–596 (1986). https://doi.org/10.1007/BF00945431
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DOI: https://doi.org/10.1007/BF00945431